On 6/30/2013 9:03 PM, William Elliot wrote: >> Nhood spaces are the DeMorgan like duals of proximity spaces.> Williard gives a closure operator cl, for a proximity space> and leaves it as an exercise to show cl is a closure operator.> The part I'm having trouble with is proving cl cl A = cl A,> which in the dual nhood space is int int A = int A.>

Look at the example in Willard concerningthe diagonal uniformities. And then lookat the definition for neighborhoods in auniformity in the previous sections.

Then for cl(A) one has

x near A implies that for everyentourage (surrounding) D there existssome t in S such that

(x,t) implies t in D[{x}]

and

(y,t) implies t in D[A]

So, D[A] fixes the values of tthat determine whether any givenx is near.